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Int. Journ. of Unconventional Computing, Vol. 1, pp. 161–177 © 2005 Old City Publishing, Inc.Reprints available directly from the publisher Published by license under the OCP Science imprint,Photocopying permitted by license only a member of the Old City Publishing Group

An Algorithm for Analyzing the TransitiveBehavior of Reversible One-Dimensional

Cellular Automata with Both Welch IndicesDifferent

JUAN CARLOS SECK TUOH MORA, MANUEL GONZÁLEZ HERNÁNDEZ

AND GILBERTO PÉREZ LECHUGA

Centro de Investigación Avanzada en Ingeniería Industrial UniversidadAutónoma del Estado de Hidalgo Ciudad Universitaria, Carr.

Pachuca-Tulancingo Km. 4.5 42184 Pachuca, Méxicojseck@uaeh.reduaeh.mx, mghdez@uaeh.reduaeh.mx, glechuga@avantel.net

Received 25 May 2004; Accepted 14 August 2004

An algorithm is presented for analyzing the transitive behavior ofreversible one-dimensional cellular automata, where reversibility refers toinjective cellular automata which have a bijective behavior for spatiallyperiodic configurations. In particular we study reversible automata wherethe ancestors of finite sequences differ at both ends (technically, with bothWelch indices different from 1). The algorithm uses the combinatorialproperties of reversible automata and their characterization by blockpermutations. A matrix presentation of the dynamical behavior andexamples are presented.

Keywords: reversible dynamics, Welch indices, block permutations, algorithms

1. INTRODUCTION

Cellular automata are dynamical systems characterized by discrete-ness in time and space. Informally, a cellular automaton is composed bythe state of the cells in a regular array, they are synchronously updatedaccording with a deterministic local interaction rule. The concept arisesby the study of John von Neumann about self-reproducing systems [17],other remarkable works are those by John H. Conway with the

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automaton “Life” [6] and by Stephen Wolfram analyzing binary one-dimensional cellular automata [19]. Cellular automata provide simplemodels of complex natural systems because they consist of a largenumber of simple interacting components which sometimes yield acomplex global behavior.

A cellular automaton is reversible if there is another local interactionrule which makes invertible the global behavior of the automaton. Theanalysis of reversible cellular automata is an interesting mathematicaltheory, and they have been applied for simulating physical and chemicalsystems [16,19], implementing data coding systems [7], and quantumcomputing [18].

However there are fundamental unsolved questions in cellularautomata theory, for instance if it is possible to know the long termbehavior of a given cellular automaton analyzing its local interactionrule. This problem remains open in the general case, but for reversibleautomata it is possible to obtain interesting results [2,1,4]. In thiscase there are two fundamental papers, the combinatorial analysis byGustav A. Hedlund [8] and their deterministic characterization usingblock permutations by Jarkko Kari [10]. Based on these works, this papergives an algorithm for analyzing the transitive behavior of reversibleone-dimensional cellular automata whose ancestors differ at both ends.

The relevance of this algorithm is that we do not need to study severalevolutions of a given reversible one-dimensional cellular automaton foranalyzing its transitive behavior, but the algorithm is able to characterizethis behavior and produce an initial configuration with an evolutionshowing the transitive features of the automaton.

The paper is organized as follows: Section 2 presents the basic con-cepts of reversible automata, their local properties and their characteriza-tion by block permutations. Section 3 explains the transitive behaviorsstudied in this work and based on the results in the previous section, weprovide a matrix representation for the dynamical behavior of reversibleautomata. Using this matrix presentation, Section 4 defines an algorithmfor detecting topologically transitive behavior and transitive points.Section 5 shows examples of these results and Section 6 gives theconcluding remarks of the paper.

2. ELEMENTS OF REVERSIBLE AUTOMATA

A one-dimensional cellular automaton is composed by a one-dimensional array of cells, every cell takes a value from a finite set K of

163ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

states and each array of states so formed is a configuration c of theautomaton. The set of all the possible configurations is presented by C.Each cell in a configuration evolves into a new state according to itscurrent state and the state of its r neighbor cells at both sides; a sequenceof 2r + 1 cells is a neighborhood, 2r + 1 is the neighborhood size and r isthe neighborhood radius. Let k represent the cardinality of K and Kn theset of sequences of n states.

A mapping j : K2r + 1 → K is the evolution rule of the automaton and itis applied synchronously to all cells in order to transform a configurationc into a new configuration c�. In this paper we shall only considerspatially periodic configurations, in this case we take periodic boundaryconditions in the sequence defining the configuration and we form a ring.Thus every cell has a well-defined neighborhood and a valid evolution,hence the initial configuration forming a ring evolves into another ring bymeans of j.

The evolution rule j induces a global mapping W between configura-tions. In this way to understanding the behavior of a cellular automatonwe must analyze the features of the evolution rule. In this paper we shalluse the term finite sequence to represent a finite sequence of states withina configuration, thus a configuration is the entire ring of the automatonand a finite sequence represents a part of this ring. Let K* be the set offinite sequences of states. For w1, w2 and w3 in K* w1w2w3 are extensions atboth sides of w2.

For w1eK*, let j (w1) be the evolution of w1 yielded applying j overeach neighborhood forming w1. If w1 and w2 belong to K* and j (w1) = w2,then w1 is an ancestor of w2. A neighborhood has 2r more cells than asingle state, so the ancestors of a sequence with n cells have n + 2r cells.For neN, j n means to apply n successive times the evolution rule over asuitable sequence weK*. For c1 and c2 spatially periodic configurationsin C, if W(c1) = c2, then c1 is an ancestor configuration of c2.

A cellular automaton is reversible if there is another evolution rule j −1

(possibly with different neighborhood size than j ) which makes invert-ible the global mapping W. Before presenting the combinatorial featuresof reversible automata,we shall explain how any cellular automaton maybe simulated by another of neighborhood size 2. In this way, we just needto study this case for understanding the general one [3,9].

2.1 Simulating any automaton by another of neighborhood size 2Consider the sequences weK4r and their evolutions in K2r. As K4r is the

cartesian product K2r × K2r, then j induces a mapping j � : K2(2r) → K2r.

164 MORA, et al.

Consider a new set S of cardinality k2r; thus the original evolution may bepresented by the mapping j � : S2 → S where every element of S repre-sents a single sequence in K2r. But j � is the evolution rule of an automa-ton of neighborhood size 2. Hence we can simulate each automaton ofany neighborhood size by another of neighborhood size 2.

For the reversible case, take between the evolution rule and its inversethe largest neighborhood size, and represent both invertible rules withthis neighborhood size. For an evolution rule with smaller neighborhoodsize, we achieve this process adding redundant states to each neighbor-hood. Once these rules have the same neighborhood size, they can bepresented by others of neighborhood size 2.

In the rest of this paper we shall only analyze reversible automataof neighborhood size 2 in both invertible rules. Now we describe theproperties of reversible one-dimensional cellular automata.

2.2 Properties of reversible automataIn the paper by Hedlund [8] two fundamental properties of reversible

automata with k states and neighborhood size 2 in both invertible rulesare proved,

Property 1. Every finite sequence has k ancestors.

Property 2. For neZ+, n ≥ 2 and weKn; the ancestors of w have L leftinitial states, a single common central part and R right final states. Thevalues L and R are known as Welch indices fulfilling that LR = k.

Thus every sequence has the same number of ancestors that all the othersand the ancestors of a given sequence just differ at the ends.

2.3 Deterministic characterization of reversible automataThe previous properties characterize the behavior of reversible

automata in a local sense; Jarkko Kari in [10] exposes how these localproperties define the global behavior in these systems as block permuta-tions. In this paper we only provide a brief explanation of this processfor reversible automata of neighborhood size 2 in both invertible rules,for more details consult references [10] and [14].

For reversible automata of neighborhood size 2 in both invertiblerules, the ancestors of each sequence begin with L distinct states andfinish with R different ones, with LR = k by Property 2. Consider all the

165ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

sequences in K3 and their evolutions in one step and define two sets Lj

and Rj as follows:

Lj = {{a, b} : aeK, beK and there is ceK such that j (a, c) = b}

Rj = {{b, c} : beK, ceK and there is aeK such that j (a, c) = b}

The elements of Lj and Rj are described in Figure 1.Thus by Properties 1 and 2, the cardinality of Lj is Lk, the cardinality

of Rj is Rk and the cardinality of the cartesian product Lj × Rj isLRk2 = k3. Then each sequence in K3 can be mapped into a single elementin Lj × Rj. Consider two sets X and Y such that the cardinality of X is Lkand the cardinality of Y is Rk; hence we can define bijections both fromLj into X and from Rj into Y.

Therefore every sequence in K3 has associated a single pair xiyj in thecartesian product X × Y where xieX, 0 ≤ i < Lk and yjeY, 0 ≤ j < Rk. Inthis way, the evolution of a reversible automaton is representedby twopermutations between blocks of three cells and a shift of one elementbetween both permutations (Fig. 2).

FIGURE 1Elements of LQ and RQ.

FIGURE 2Evolution of a reversible automaton represented by block permutations and a shift.

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Based both on the local properties and the characterization by blockpermutations of reversible automata, we shall present an algorithm forcalculating transitive configurations and topologically transitive behav-iors in these systems. In the next section we expose the formal definitionsabout transitive properties as well as the matrix presentation which thealgorithm is based on.

3. DYNAMICS OF REVERSIBLE AUTOMATA

In order to study the dynamical behavior of reversible automata, weshall give special attention in handling configurations. For a configura-tion ceC and ieZ, c[i] represents the cell at position i; thus, for i, jeZwith i ≤ j, c[i,j] is the sequence of states between positions i, j in c. For ieNand weK2i+1, a cylinder set C[w] is the set of configurations such that eachceC[w] fulfills that c[−i,i ] = w.

For w1 and w2 in K2i+1, the cylinder sets C[w1] and C[w2] are disjoint if w1 ≠w2, because each configuration c1eC[w1] is different from all configurationc2eC[w2]. For K2i+1, CK2i+1 is the family of cylinder sets C[w] for every weK2i+1.Therefore CK2i +1 is the whole set of configurations and it defines a partitionof C.

In the reversible case, the global mapping W between configurationsinduced by the evolution rule j is invertible, therefore these systems areinvertible dynamical systems. Special features of the dynamical behaviorare defined depending on the transitions between the cylinder sets ofthe automaton; in particular we shall take two definitions about the tran-sitive behavior in reversible automata. These ones are based on thoseexposed in [12,15,5].

Definition 1. A transitive point in a reversible automaton is a configurationwhich visits all the cylinder sets in CK2i+1 by means of the global mapping.

Definition 2. A reversible automaton is topologically transitive if each cyl-inder set has a configuration whose evolution visits all the other cylindersets applying the global mapping.

With the representation of the reversible behavior by block permuta-tions, we may specify the set of cylinder sets intersected by the evolutionof configurations in another one. In this paper we shall only study thefamily CK3 but the analysis can be extended for larger families. For weK3,

167ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

concatenate w with all the possible sequences in K3 at both sides. Thus forwL, wR elements of K3, sequences of the form wLwwReK9 are produced.

Let us apply twice the block permutations in each sequence of K9,which yields a new sequence ueK3. In this way, we can define a transition(or a two-step evolution) from a cylinder set C[w] to distinct cylinder setsC[u] placed at the same coordinates (Fig. 3).

We present these transitions by a matrix Mj; in this matrix, row andcolumn indices are all the cylinder sets in CK3. For every w1 and w2 in K3,each entry (C[w1], C[w2]) in Mj is defined as follows:

( , )[ ] [ ][ ] [ ]

C CC C

w ww w

1 2

1 21

0=

if there is a transition from into

otherrwise

(1)

Thus Mj is a 0 − 1 matrix representing the one-step transitions amongthe sets of CK3. Using this matrix as basis, we shall present an algorithmfor calculating transitive configurations and analyzing the transitivebehavior of a given reversible automaton.

4. TRANSITIVE BEHAVIOR AMONG CYLINDER SETS

Let C[w1] → C[w2] mean that there exists neZZ+ such that there is aconfiguration in C[w1] whose evolution in 2n steps produces a

FIGURE 3Transitions from the set C[w] to distinct sets C[u].

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configuration in C[w2]. Note that (C[w1], C[w2]) = 1 in Mj indicates thatC[w1] → C[w2]. For reversible automata with both Welch indices differentfrom 1, the extensions of one cell of the ancestors cannot evolve in all thepossible states of K. Therefore if C[w1] → C[w2] and C[w2] → C[w3], it does notimply that C[w1] → C[w2] → C[w3].

Nevertheless, the matrix Mj is useful as basis for knowing the possibletransitions between the cylinder sets, in order to resolve if the transitionC[w1] → C[w2] → C[w3] exists, we need a better analysis of the extensions of theancestors. First of all, we present a result about the symmetric behaviorof the transitions among cylinder sets.

Lemma 1. For any pair of sequences w1, w2 elements of K3, if C[w1] → C[w2]

then C[w2] → C[w1].

Proof. If C[w1] → C[w2] then for some neZ+ there are sequences a, b in Kn

such that j 2n(aw1b) = w2. Consider the spatially periodic configurationc = · · ·aw1baw1baw1b· · ·. Then, W2n(c) = c� where c� is also a spatially peri-odic configuration with form · · ·cw2dcw2dcw2d· · · with c, d elements of Kn.But cw2d is a finite sequence, so the evolution of c� can only generate afinite number of different configurations, forming a periodic evolution.Since the automaton is reversible, then the evolution returns to c� but alsoreturns to c. Thus, c is both ancestor and successor of c� and thereforeC[w2] → C[w1]. a

From the proof of Lemma 1 we obtain another implication.

Corollary 1. The transition between cylinder sets is a symmetric and areflexive relation.

Suppose that there are transitions C[w1] → C[w2] and C[w2] → C[w3] in Mj.Based on these ones, we define now a procedure for reviewing if thereexists the transition C[w1] → C[w2] → C[w3]:

Algorithm 1 (Detecting a transition between cylinder sets).

1. Consider the right ending cells from w1, w2 and the intermediateevolution (the set of states in these cells are labeled F1, F2 and F3

respectively in Fig. 4).2. For F1, take all the right successor states specified by j ; and for F2,

take the right successor states defined by j −1. Choose the common

169ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

states in both cases, this new set of states shall be represented by F4

(Fig. 4).3. If F4 = Ø then the transition is not possible and the algorithm stops.

Otherwise, verify all the combinations of F3 with all the cells in F4

and choose the pairs yielding states in both F2 by j and F1 by j −1.4. If some state in F3 or F4 is not used for generating some state in F1

and F2, then it is discarded from the set. If some state in F1 or F2 isnot generated by any combination of states in F3 and F4, then it isdiscarded from the set.

5. The same process is applied for all the ending cells in the transitionof Fig. 4. Applying iteratively this process, if the set of states in theending cells at both sides are not empty, then there is a sequenceveK5 such that w2 is its central part and v specifies a spatiallyperiodic configuration c = · · ·vvv· · · whose evolution holds thatC[w1] → C[w2] → C[w3] by Lemma 1 (Fig. 5).

FIGURE 4Cells in the right superior corner of the transition. F4 is the intersection of the successorstates of F1 and F2 applying j and j −1 respectively.

FIGURE 5If the sets of states in the gray cells are not empty, then C[w1]

→ C[w2]→ C[w3].

170 MORA, et al.

In a first glance, the previous algorithm uses only the results of Hedlundto calculate the transitions between cylinder sets, so one question is howthe block permutations are related with Algorithm 1. The answer is intwo aspects, the first one is that we are taking the matrix Mj as referencefor analyzing if there is a transition from C[w1] into C[w3], so the block per-mutations determines the transitions to analyze. The second point is inthe length of the sequences taken for studying the transitions between thecylinder sets, this length is the one required for the block permutations,thus if the algorithm finds a transitions from C[w1] into C[w3], then there is acomposition of block permutations taking w1 into w3 and therefore thereis a configuration in C[w1] whose evolution yields a configuration in C[w3].

Algorithm 1 depends on the extensions of the ancestors defined byboth invertible rules, so this process does not depend of the number ofsequences weK taken for reviewing the transitions between cylinder sets;if we take a larger number of sequences we just need to apply more itera-tions of both invertible rules j and j −1 to obtain the possible transitionsbetween these sequences. In this way we can revise larger transitionsbetween cylinder sets; for selecting which transitions shall be analyzed,we take the matrix Mj as basis, powers of this matrix specifies possibletransitions in several steps from one cylinder set to another (notice thatnot all the transitions specified by a power of Mj are correct due to thevalues of the Welch indices).

For neZ+, we can analyze a transition C[w1] → C[w2] → · · · → C[wn] speci-fied by the n-th power of Mj. Algorithm 1 can review it and detect if it isa valid transition, that is, if there exists an intermediate sequence veK2n

such that it defines a spatially periodic configuration whose evolutionpasses through the cylinder sets in this transition. Based on this algorithmwe can define a new one for calculating the transitions among all the cyl-inder sets of the reversible automaton, obtaining as well relevant featuresabout its dynamical behavior. This algorithm updates the entries of Mj

as a valid transition is calculated.

Algorithm 2 (Transitions among cylinder sets).

1. For the transition C[w1] → C[w2] → · · · → C[wn], if C[w1] → C[w2] → · · ·→ C[wn−1] is a valid transition and (C[w1], C[wn]) = 0 in Mj, then Algo-rithm 1 is applied for reviewing the transition C[w1] → C[w2] → · · ·→ C[wn].

171ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

2. If C[w1] → C[w2] → · · · → C[wn] is a valid transition, then we update(C[w1], C[wn]) = 1 in Mj and now the successor sets of C[wn] are studied instep 1. Otherwise C[wn] is rejected from the current transition.

If we have a valid transition defined by several cylinder sets, Algorithm 2verifies a larger transition adding the successor sets of the last one in thetransition. If a given sequence of cylinder sets does not describe a validtransition, then the last set is eliminated and the successors from the nowlast set are analyzed. We said before that if there is a transition C[w1] → C[w2]

and another one C[w2] → C[w3] then it is not always true that C[w1] → C[w3],however Algorithm 2 certainly defines a transitive relation among thecylinder sets of the automaton.

Lemma 2. Algorithm 2 defines a transitive relation among the cylinder sets.

Proof. For neZ+, if C[w1] → C[w2] → · · · → C[wn] is a valid transition thenAlgorithm 2 calculates a sequence veK2n defining a spatially periodicconfiguration whose evolution passes through all the cylinder setsin this transition. Therefore for 1 ≤ n1 < n2 < n3 ≤ n if C C[ ] [ ]w wn n1 2

d andC C[ ] [ ]w wn n2 3

d , then C C[ ] [ ]w wn n1 3d by the evolution of the periodic

configuration. a

Using Corollary 1 and Lemma 2 we obtain the next result:

Corollary 2. Algorithm 2 defines an equivalence relation in Mj.

The proof of Lemma 2 indicates that for each class in this equivalencerelation there exists a periodic configuration passing through all thecylinder sets in the class. For sequences defining valid transitions, Mj isupdated in Algorithm 2 showing the transitive behavior between thecylinder sets of the automaton for large transitions. This matrix has thefollowing graph representation: if the entry (C[wi]

, C[wj]) = 1 then it is filled

by black, otherwise it is not filled.A problem with this process is the number of possible transitions to

analyze for knowing the features of the transitive behavior. However ifthere is a transition connecting a cylinder set with another one, then byLemma 1 there exists another transition going from the final cylinderset to the initial one. Thus we just need to study the transitions of onecylinder set for reviewing the transitions of other ones.

If Algorithm 2 yields that Mj has a row with only positive entries, thismeans that the cylinder set represented by this row has a valid transition

172 MORA, et al.

to all the others and by Lemma 1, there are valid transitions from eachcylinder set to all the rest. In this way all the cylinder sets are connectedone another, and the automaton is topologically transitive. We can getenough information to yield a transitive point; for a row in Mj with justpositive entries produced by Algorithm 2 and the sequence formed byall the cylinder sets defining a valid transition, we apply the followingalgorithm:

Algorithm 2 (Calculating a transitive point).

1. Apply Algorithm 1 to this sequence of cylinder sets to verify if itdefines a valid transition (which is true).

2. Keep in each step of Algorithm 1 the sequence of states formed inthe middle of the transition.

This central sequence specifies a spatially periodic configuration passingthrough all the cylinder sets in the transition by Lemma 1 and thereforedefining a transitive point.

On the other hand, Algorithm 2 may generate a row which doesnot have a transition connecting it with all the other cylinder sets, evenreviewing all the possible transitions. Hence this algorithm is also usefulfor detecting reversible automata where not all the cylinder sets are con-nected; that is, reversible automata which are not topologically transi-tive. These automata are characterized by forming membranes andmacrocells in their evolutions [13,19]. In the next section we shall showexamples of these transitive behaviors and how Algorithm 2 is able todetect them.

5. EXAMPLES

In this section, two reversible automata of 4 states and neighborhoodsize 2 are presented, that is, the simplest case where both Welch indicesare different from 1. The first example is topologically transitive; that is,there exists a valid transition from one cylinder set to any other. Thesecond example does not show such a behavior. Consider the automatonin Fig. 6-A, the evolution rule is presented by a matrix; the rowand column indices are the left and right cells respectively of theneighborhoods and each element of the matrix is the evolution of theneighborhood represented by its coordinates.

173ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

FIGURE 6Reversible automata of 4 states and neighborhood size 2.

Algorithm 2 yields the matrix Mj represented in Fig. 7-A. In this casethere is a valid transition from each cylinder set to all the others, thereforethe automaton is topologically transitive.

For the automaton in Fig. 6-A, Algorithm 2 yields a spatially periodicconfiguration with a transitive behavior formed by the repetition ofthe sequence 0102002300130120200033001120013111102130301230332-12312312122003221100331222122130313233313323202323 (we do notshow the evolution of this sequence due to its impractical graphpresentation).

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FIGURE 7Graph presentation of the matrices Mj.

Consider now the automaton in Fig. 6-B, in this example the evolutionis characterized by forming membranes and macrocells. The graphrepresentation of the matrix Mj generated by Algorithm 2 is in Fig. 7-B,we can see that no cylinder set is connected with all the others, showingthe particular behavior of the automaton. Algorithm 2 also indicatesthat in this example Mj defines an equivalence relation formed by severalclasses, where each class is a set of cylinder sets specifying a validtransition. Thus applying Algorithm 2 we can find a spatially periodicconfiguration whose evolution pass through all the cylinder sets in theclass.

Using this process for the automaton in Fig. 6-B for the class contain-ing the set C[000], a configuration passing for all the cylinder sets of thisclass is obtained by repetitions of the sequence 0212222000 (Fig 8).

6. CONCLUSIONS

The characterization of a reversible automaton by means of block per-mutations over cylinder sets provides a suitable matrix representationuseful as a basis for analyzing its dynamical behavior. This paper haspresented an algorithm using the extensions of the ancestors specified byj and j −1 to calculate the transitions among cylinder sets. With thiswe obtain useful information about the transitive behavior of a given

175ANALYSING BEHAVIOUR OF REVERSIBLE CELLULAR AUTOMATA

reversible one-dimensional cellular automaton, such as defining if theautomaton is topologically transitive.

The algorithm presented in this paper is a better option than the naiveapproach of considering several configurations and their evolutions forstudying the transitive behavior of a reversible automaton. With thisalgorithm we do not need to evolve several configurations to find out thetransitions between cylinder sets, but we can generate a single initialconfiguration whose evolution passes through a predefined family ofcylinder sets, obtaining an example of the dynamical behavior.

The restriction of using periodic boundary conditions in the spatialconfigurations does not rest generality to the algorithm since any cellularautomata is transitive if and only if it is transitive on spatially periodicconfigurations using the classical Cantor topology [11].

The reason of using blocks of size 3 for defining the cylinder sets is toprovide an easier analysis, but it is not essential for constructing Mj.Larger blocks yield a more detailed study but also an algorithm requiringmore computational resources.

FIGURE 8Evolution of the spatially periodic configuration formed by 0212222000. This evolutionpasses through all the cylinder sets in the class containing C[000].

176 MORA, et al.

A better dynamical analysis is necessary because the processespresented in this paper allow to analyze a few features of the transitivebehavior of a reversible automaton. Another way of calculating morecharacteristics of a reversible automaton is desirable, for instance todetect topologically mixing behaviors.

Suggested works consist of obtaining a deterministic classification forthe dynamical behavior of these systems, and the adequate use of theblock permutations for controlling the evolution of a given automaton,that is, controlling the transition from a cylinder set to another for a fixednumber of steps.

AcknowledgmentsIn particular to Enrico Formenti for his aid and useful comments in

order to improve both the theoretical basis and presentation of the paper.

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